component displacements being in opposite directions, the total displacement is equal to KQ – Kq, and since KQ is equal to Kq, the displacement is zero. Hence if we construct a curve such that the ordinates are everywhere equal to the algebraic sum of the ordinates of the two component curves, this curve will represent the resultant displacement. The resultant thus obtained is shown dotted in Fig. 42.

In Fig. 43 the same curves are compounded, but the time scale is made smaller, so that more periods of each curve may be shown. It will be seen that the resultant curve, although not a sine curve, is a periodic

A

B

FIG. 43.

curve, and hence the resultant motion is periodic, the period being equal to AB, i.e. to five times the period of the quicker vibration, or four times that of the slower.

If the two S.H.M.'s to be compounded are of nearly the same period, say in the ratio of 9: 10, then the compound harmonic curve obtained will, as shown in Fig. 44, everywhere approximate to the form of a sine curve, but the amplitude will alternately wax and wane; the maxima occurring when the component vibrations are exactly in phase, and the

FIG. 44.

minima when the phases differ by half a period. As in 9 periods of the slower vibration there occur 10 periods of the quicker, in this interval one will have gained exactly one period on the other, and they will again be in the same phase. Thus the curve shows that the maxima occur at every 10th period of the quicker vibration.

This waxing and waning of the resultant motion, when two S.H.M.'s of nearly the same period are compounded, is the cause of the phenomenon of beats in music, and will be further studied when we come to the subject of sound.

55. Fourier's Theorem.-In the previous section we have compounded two harmonic curves and drawn a resultant curve. The same method can be employed to compound any number of harmonic curves. The curves having all been drawn, with their appropriate amplitude,

period, and phase, the resultant curve is drawn so that at every point its ordinate is equal to the algebraic sum of the ordinates of the component curves at that point. By suitably choosing the period and amplitude of

the component harmonic curves, it is possible, as illustrated in Figs. 45 and 46, to produce a periodic resultant curve of a type very different from a sine curve.

Fourier first showed that any periodic curve, as long as it nowhere goes to an infinite distance from the axis of X, can be built up by compounding together a finite number of harmonic curves the periods of which are commensurate. This last condition is necessary, for otherwise the resultant curve obtained by compounding the curves would never

exactly repeat itself, and would not be periodic. Hence it follows that any periodic motion can be considered as the resultant of a number of commensurate S.H.M.'s. If T is the period of the complex periodic motion, then the periods of the component S.H.M.'s will be included in the numbers T, T2, T3, T4, &c.

As an illustration of the way in which a periodic curve of a given form may be built up by the combination of a number of S.H.M.'s, suppose the required curve to be represented by the lines ABCDEFG (Fig. 46). The thick continuous curve given in the figure is obtained by compounding the three S.H.M.'s shown dotted, of which the frequencies are in the ratio 1:3:5, while the amplitudes are as 1: 1/31/5. It will be seen that even with three terms an approximation to the required form is produced. In Fig. 47 the result of combining 100 S.H.M.'s, having frequencies

FIG. 47.

proportional to the numbers 1, 3, 5, 7, 9, &c., and amplitudes proportional to 1, 1'3, 15, 1/7, 1/9, &c., is shown on a reduced scale. It will be noticed that in this case the required curve is almost perfectly reproduced.

Machines have been devised, called harmonic analysers, to determine mechanically the amplitudes of the S. H. M.'s of the periods T, 7/2, T/3, &c., required to build up any given curve. Other machines are capable of drawing the resultant of a certain number of S. H. M.'s of given amplitude and period.

E

PART III-DYNAMICS

CHAPTER VIII

NEWTON'S LAWS OF MOTION

56. Subdivisions of Dynamics.-Up to the present the motion of bodies has been considered quite in the abstract, and although we have assumed that the motion varied in certain ways, we have not inquired into the causes of these variations. We now pass on to consider the effects of force as shown in its action on the motion or equilibrium of material bodies. This branch of the subject of mechanics is called Dynamics. Dynamics is sometimes subdivided into two sections; in one, called Kinetics, the effect of forces on the motion of bodies is studied, while in the other, called Statics, the conditions which must exist if a body remains at rest when acted upon by a system of forces are investigated.

57. Stress. When one portion of matter acts on another portion, so as to influence its state, then the whole phenomenon of the mutual action of the two portions of matter is called in general a stress. In certain particular cases the stress has received a special name; thus we have a tension, a pressure, a torsion, an attraction, a repulsion, &c.

The term stress includes the consideration of both the mutually influencing portions of matter; it is, however, sometimes useful to concentrate our attention on one aspect of a stress, namely, the action on one of the portions of matter, so that we regard the stress as something acting on this piece of matter. From this point of view we say that the phenomena which we observe are the effect of External or Impressed Force on the portion of matter in question, and are due to the ACTION of the other portion of matter. The opposite aspect of the same stress would in this case be called the reaction on the other portion of matter. Hence Action and Reaction are simply different aspects of a stress, just as buying and selling are different aspects of one and the same transaction, according as we look at it from the point of view of one or other of the persons taking part in the transaction.

58. Newton's Laws of Motion.-The effect of external or, as it is sometimes called, impressed force on the motion of bodies is defined in three laws which are known as Newton's Laws of Motion. The first of these laws deals with the behaviour of a body when no external force

acts on it. The second tells us how the external force, when acting, may be measured. The third compares the two aspects of a stress, namely, Action and Reaction. These laws are Axioms, and do not admit of direct experimental proof; they depend, however, on convictions drawn from experiment, and their truth is universally admitted by those who have sufficient physical knowledge to thoroughly understand their purport.

59. Newton's First Law." Every body continues in its state of rest or of uniform motion in a straight line, unless it be compelled by impressed force to change that state."1

This law is also known as the law of Inertia, since it states that no body is capable of altering its state of rest or of motion without the intervention of some outside influence; and this fact we express in scientific language by saying that every body has inertia.

The law in the first place gives a definition of force, since it states that force is that action by means of which the state of rest or motion of a body is changed, and that unless a force acts no such change will occur. We may therefore define force as that which tends to produce change of motion in a body on which it acts.

In the next place the law tells us how a body will move when it is unacted upon by external forces. It says that if the body is in motion then it will continue moving uniformly in a straight line, if at rest it will continue at rest.

Indirectly the law may be taken as defining equal times. The times which a body, unacted upon by external forces, takes to pass through equal spaces are equal.

Since we are unable to obtain a body which is entirely unacted upon by external force, we cannot experimentally prove that if once set in motion it would continue to move uniformly. We find, however, that the more we reduce the magnitude of the impressed forces acting on a body, the greater is its tendency to continue moving at a uniform rate in a straight line when once it has been set in motion. Thus we know that if a stone is thrown along the surface of a road it will soon lose its motion. If thrown along the surface of smooth ice-in which case the friction, which is an impressed force tending to check the motion, is much less than in the case of the road-it will, however, continue to move very much longer.

A much more powerful argument for the validity of the law is obtained by considering that we can by its means solve problems in mechanics, and the solutions thus obtained always agree with observation, so that we conclude that our fundamental assumption is correct. Thus every one who makes use of the Nautical Almanack to discover the position of a star or the time of an eclipse, tacitly allows the correctness of Newton's law, for it is by the assumption of the correctness of the law that the numbers there given have been calculated.

1 Corpus omne perseverare in statu suo quiescendi vel movendi uniformiter in directum, nisi quatenus illud a viribus impressis cogitur statum suum mutare.

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